# Orthogonal projection operator onto the distriubution (and its complement) of tangent space of a Riemannian manifold (geometric mechanics)

Suppose $$Q = \mathbb{R}^2\times S^1\times S^1$$ be the configuration space for a rolling disk with a Riemannian metric, $$\mathbb{G}$$, representing the kinetic energy at each configuration whose matrix representation is

$$[\mathbb{G}] = \begin{bmatrix}m&0&0&0\\ 0&m&0&0\\ 0&0&J&0\\ 0&0&0&I \end{bmatrix}$$ with respect to coordinates $$q = (x,y,\theta,\phi)$$.

The nonholonomic constraints are specified by

\begin{align} & \dot{x} = \rho\cos\theta \dot{\phi}\\ & \dot{y} = \rho\sin\theta \dot{\phi}. \end{align}

In matrix form, the constraints are specified by

\begin{align} \underbrace{\begin{bmatrix}1&0&0&-\rho\cos\theta\\ 0&1&0&-\rho\sin\theta \end{bmatrix}}_{\Omega(q)}\begin{bmatrix}\dot{x}\\ \dot{y}\\ \dot{\theta}\\ \dot{\phi} \end{bmatrix} = \begin{bmatrix}0\\0 \end{bmatrix}. \end{align}

The constraint distribution, $$\mathcal{D}_q \subset T_qQ$$, can be specified by

\begin{align} G(q) = \begin{bmatrix}\rho\cos\theta &0\\ \rho\sin\theta&0\\ 0&1\\ 1&0 \end{bmatrix} \end{align}

and the complement of the constraint distribution $$\mathcal{D}_q^\bot$$ can be specified by

\begin{align} \Omega^T = \begin{bmatrix} 1&0\\ 0&1\\ 0&0\\ -\rho\cos\theta&-\rho\sin\theta \end{bmatrix} \end{align}

I am interested in constructing orthogonal projections onto the constraint distribution and its complement

\begin{align} & \mathcal{P}_{\mathcal{D}}:TQ\rightarrow TQ\\ & \mathcal{P}_{\mathcal{D}^\bot}:TQ\rightarrow TQ\\ \end{align} but I'm not sure how do it.

In this thesis Control of nonholonomic mechanical systems using virtual surfaces (page 45), the author proposes the following matrix construction:

\begin{align} & [\mathcal{P}_{\mathcal{D}}] = [G]\left([G]^T[\mathbb{G}]G \right)^{-1}[G]^T[\mathbb{G}]\\ & [\mathcal{P}_{\mathcal{D}^\bot}] = [\mathbb{G}]\Omega^T\left([\mathbb{G}]^{-1}\Omega[\mathbb{G}][\mathbb{G}]^{-1}\Omega^T \right)^{-1}[\mathbb{G}]^{-1}\Omega[\mathbb{G}] \end{align} Personally, I have no idea how the author came up with these maps (the author also admits there are multiple ways to do this). Can anyone offer any suggestions how to construct such projection maps??

• Look at any standard linear algebra book for the formula for projection onto the column space of a matrix $A$ with maximal column rank: It is $P=A(A^\top A)^{-1}A^\top$. Commented Jul 19, 2021 at 1:39